I didn't realize this until I took my first trip on the 744. The wings curve downward at the last few feet on each wing. Also, if you notice sitting in front of the wing, the last 10-20 feet of the leading edges have a different angle than the rest of the wing. What are the purposes of these features?

I'm not talking about the entire wing flexing due to fuel. I know that the entire wing will flex because of the weight of fuel. I'm talking about how the very tip of the wing, close to the winglet has an extra downward bend to it. Even when the wing flexes upward during flight, there is a bend in the tips.

I would guess it has something to do with the fact that the outermost portion of the wing is actually an extension "tacked" onto the original wing design. IIRC, the -4XX 747s have an extra two metres of span (per wing), in addition to the winglets themselves.
To me, it appears that (in flight at least), that last outer portion is actually even (0 degrees to level), whereas the rest of the wing has a dihedral form, giving the illusion of a "droop."

The section of the wings outboard of the No. 1 and No. 4 engines do indeed have a slightly reduced sweepback angle. The purpose of that is to reduce wing flutter. I read about that in a book (forgot the name) that documents the development of the 747.

Actually the degree of the leading edge angle of all the 747 wings is higher up to the outer engine joint. The reason is aerodynamics. I´m not a technician, but that´s for sure the reason for that. The A380 will be similar. The leading edge angle degree near the fuselage will be higher than on the outer parts.